Xjyuk

When learning about relativity, it is said that velocity (an object or a particle traveling close to the speed of light) results in time moving slower for the object moving.

However, there is also gravitational time dilation and the mass-energy equivalence as given by E^2 = (pc)^2 + (m0C^2)^2 which takes momentum into account.

So what is the true cause of time dilation? Is it is velocity? Momentum? Perhaps energy?

As an example, let's say we have a metallic ball with a lot of momentum. It has so much momentum that it would travel at 10% the speed of light. But it is restricted by an equally powerful magnet so that it does not move at all. Would that system have time dilation? My understanding is that it would because there is so much energy in the system, so it would warp spacetime.

Time dilation can be caused by as you mention relative speed (as per SR) or gravitational time dilation (as per GR).

Please see from wiki:

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field.

1. Relative speed SR

SR indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to him, will be measured to tick slower relative to the clock that is at rest.

2. Gravitational time dilation GR

Please see from wiki:

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass.

It is a misunderstanding that (gravitational) time dilation is caused by mass. In reality it is caused by stress-energy, and the difference between stress-energy of two points in space. For example, clocks near the Sun seem to tick slower compared to clocks here on Earth, because of the difference between the stress-energy of the Sun and Earth. A very good example for this is the Shapiro effect. Please see here:

https://en.wikipedia.org/wiki/Shapiro_time_delay

You are correct to say that, anything that has stress-energy will warp spacetime, in fact even a photon, having stress-energy will warp spacetime, and will have its own gravitational effects.

In your example, the metallic ball will have its own stress-energy and its own gravitational effects and yes it will warp spacetime. Yes, there would be time dilation between a clock at the metallic ball, and a clock far away from it in empty space. This time dilation would be caused by the difference in stress-energy between the metallic ball and empty space.